How does Peskin derive equations 2.51 and 2.52 from equation 2.50 on page 27?

[Rapivcci]

I am trying to verify 2.50 on pg 27 of peskin but did it in my own way, I am sure i made some mistakes here but i was able to get the right answer. Can someone highlight why some of these steps are invalid and explain how peskin git from 2.50 to the first step used in the text for 2.51.

And yes it should be p-m(p+m), sry about that.

 

[strangerep]

how peskin git from 2.50 to the first step used in the text for 2.51.

From (2.50), with the assumption that x-y is purely timelike, that reduces the exponent in (2.50) to $-iEt$, which is $-it\sqrt{p^2+m^2}$, (where $p^2$ means the magnitude of the 3-momentum).

Then he changes from Cartesian momentum coordinates $p_x, p_y, p_z$ to spherical polar momentum coordinates $p_r, p_\theta, p_\phi$. The Cartesian $p^2$ is just $p_r^2$, but then he denotes a plain $p$ to mean $\sqrt{p_r^2}$, i.e., the magnitude of the 3-momentum. The integral in the 1st line of (2.51) is then really an integral over $p_r$ (analogous to an integral over $r$ in ordinary spherical polar position coordinates). He has already performed the angular integrals of the 3-integral in (2.50) -- that's where the leading factor of $4\pi$ in (2.51) came from.

Then there's a change of integration variable from $p$ to $E=\sqrt{p^2+m^2}$ to get the 2nd line in (2.51).

HTH.

 

[Rapivcci]

@strangerep it helps a lot, thanks a bunch.